We prove that any finite collection of polygons of equal area has a
common hinged dissection. That is, for any such collection
of polygons there exists a chain of polygons
hinged at vertices that can be folded in the plane continuously
without self-intersection to form any polygon
in the collection. This result settles the open problem about
the existence of hinged dissections between pairs of polygons that
goes back implicitly to 1864 and has been studied extensively in the
past ten years. Our result generalizes and indeed builds upon the
result from 1814 that polygons have common dissections (without
hinges).
Our proofs are constructive, giving explicit algorithms in all
cases. For two planar polygons whose vertices lie on a rational grid,
both the number of pieces and the running time required by our construction
are pseudopolynomial.
This bound is the best possible, even for unhinged dissections.
Hinged dissections have possible applications to reconfigurable
robotics, programmable matter, and nanomanufacturing.